Simplifying Square Roots

To simplify a square root: make the number inside the square root as small equally possible (but still a whole number):

Instance: √12 is simpler as ii√3

Get your reckoner and check if you desire: they are both the aforementioned value!

Here is the dominion: when a and b are non negative

√(ab) = √a × √b

And here is how to utilize it:

Instance: simplify √12

12 is 4 times 3:

√12 = √(4 × iii)

Utilize the rule:

√(4 × 3) = √4 × √three

And the square root of 4 is ii:

√4 × √3 = 2√three

So √12 is simpler as 2√3

Some other example:

Instance: simplify √viii

√8 = √(4×2) = √iv × √2 = 2√ii

(Because the foursquare root of 4 is 2)

And some other:

Instance: simplify √eighteen

√18 = √(ix × 2) = √9 × √2 = 3√two

It often helps to factor the numbers (into prime numbers is best):

Case: simplify √6 × √15

First we can combine the two numbers:

√vi × √15 = √(6 × 15)

Then we factor them:

√(half-dozen × fifteen) = √(2 × 3 × iii × five)

Then we see two 3s, and determine to "pull them out":

√(2 × 3 × 3 × 5) = √(iii × 3) × √(two × 5) = 3√x

Fractions

In that location is a similar rule for fractions:

root a / root b = root (a / b)

Example: simplify √30 / √x

Start we tin can combine the two numbers:

√thirty / √10 = √(30 / ten)

Then simplify:

√(30 / 10) = √3

Some Harder Examples

Example: simplify √20 × √5 √two

See if yous can follow the steps:

√20 × √five √ii

√(two × 2 × v) × √five √2

√2 × √2 × √5 × √v √2

√ii × √5 × √5

√2 × 5

5√2

Example: simplify ii√12 + 9√3

First simplify 2√12:

2√12 = 2 × two√3 = 4√three

Now both terms take √three, we can add them:

4√three + 9√iii = (4+9)√3 = thirteen√iii

Surds

Notation: a root we tin't simplify further is called a Surd. So √3 is a surd. But √four = 2 is non a surd.